You need to connect the GND of the two boards together, otherwise no communication can happen.
The default Jetson Nano UART and I2C level is 3.3V, so if you are going from the Jetson Nano carrier board pins to the Arduino pins, you should go from 3.3V to 5.0V. (I’m assuming you’re using an old-school AVR based Arduino, not the newer ARM based ones, which are just play 3.3V.)
Or, it turns out, the Arduino is pretty lenient with voltages (because the AVR chip can also run at 3.3V, or even 2.5V, depending on what you need) so it may detect 1 and 0 just fine with a straight 3.3V signal. Thus, you can use a resistive divider from the Arduino Tx to the Jetson Rx (use 2.2 kOhm from Arduino Tx to Jetson Rx, then 4.7 kOhm from Jetson Rx to ground) and just wire Jetson Tx straight to Arduino Rx. Just don’t turn on pin pull-up for the Rx pin on the Arduino! (And beware if you also use the serial port to program it, in which case that may pull the pin too a too high voltage.)
Another option is to use the Arduino Nano Pro 3.3V / 8 MHz, and just run everything at 3.3V.
But, that being said – the TXB0108 should work fine, if you use the GPIO pins, and hook the high voltage part to 5.0V, the low voltage part to 3.3V, and make sure to tie the grounds together between the boards.